Binomial theorem

Tell me about the binomial theorem!

Binomial theorem — the short, useful version

For any nonnegative integer $n$ and any numbers $a,b$,

$$ (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}\,a^{\,n-k}b^{\,k}, $$

where the binomial coefficients are

$$ \binom{n}{k}=\frac{n!}{k!(n-k)!}\qquad(0\le k\le n). $$

So you expand $(a+b)^n$ as a sum of terms $a^{n-k}b^k$ weighted by $\binom{n}{k}$.

Small examples

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$

$$ (a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 $$

(For the middle coefficient in the last line: $\binom{4}{2} = \dfrac{4!}{2!2!}=\dfrac{24}{4}=6$.)

Two quick ways to think about the coefficients

• Combinatorial: $\binom{n}{k}$ counts how many ways to choose $k$ of the $n$ factors to contribute a $b$ (the others contribute $a$).
• Pascal’s identity: $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$. This builds Pascal’s triangle, whose rows are the coefficient sequences.

Proof sketches

• Combinatorial proof: When you multiply $(a+b)(a+b)\cdots(a+b)$ $n$ times, each term in the expansion picks either $a$ or $b$ from each factor. A term with exactly $k$ $b$'s has exponent pattern $a^{n-k}b^k$, and there are $\binom{n}{k}$ such choices.
• Induction: Known for $n$; multiply by $(a+b)$ and collect like terms, using Pascal’s identity to get the coefficients for $n+1$.

Generalized binomial theorem (non-integer exponents)

For arbitrary real or complex $\alpha$,

$$ (1+x)^{\alpha}=\sum_{k=0}^{\infty}\binom{\alpha}{k}x^k \quad\text{where}\quad \binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}. $$

This infinite series converges for $|x|<1$ (unless $\alpha$ is a nonnegative integer, in which case it truncates to the finite polynomial above). Example:

$$ (1+x)^{1/2}=1+\tfrac12 x - \tfrac18 x^2 + \tfrac{1}{16} x^3 +\cdots\qquad (|x|<1). $$

Applications

• Algebraic expansions and simplification.
• Combinatorics and counting.
• Probability: coefficients appear in the binomial distribution $\Pr(\text{exactly }k)=\binom{n}{k}p^k(1-p)^{n-k}$.
• Calculus / series: Taylor expansions (Newton’s generalized binomial theorem).
• Generating functions, identities, and number theory.

Handy identities

• Sum of row: $\displaystyle\sum_{k=0}^n \binom{n}{k}=2^n$ (plug $a=b=1$).
• Alternating sum: $\displaystyle\sum_{k=0}^n (-1)^k\binom{n}{k}=0$ for $n\ge1$ (plug $a=1,b=-1$).
• Vandermonde’s identity (useful convolution): $\displaystyle\sum_{k} \binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}$.

Would you like a full induction proof written out, a visualization of Pascal’s triangle, some practice problems (with solutions), or an example applying the generalized theorem (e.g., expand $(1+x)^{1/2}$ to 5 terms)?